Schrodinger wave equation systems

Just like any spectroscopic event EPR is a quantum-mechanical phenomenon, therefore its description requires formalisms from quantum mechanics. The energy levels of a static molecular system (e.g., a metalloprotein in a static magnetic field) are described by the time-independent Schrodinger wave equation,... 

A rigorous mathematical formalism of chemical bonding is possible only through the quantum mechanical treatment of molecules. However, obtaining analytical solutions for the Schrodinger wave equation is not possible even for the simplest systems with more than one electron and as a result attempts have been made to obtain approximate solutions a series of approximations have been introduced. As a first step, the Bom-Oppenheimer approximation has been invoked, which allows us to treat the electronic and nuclear motions separately. In solving the electronic part, mainly two formalisms, VB and molecular orbital (MO), have been in use and they are described below. Both are wave function-based methods. The wave function T is the fundamental descriptor in quantum mechanics but it is not physically measurable. The squared value of the wave function T 2dT represents probability of finding an electron in the volume element dr. 

The coefficients of the various p orbitals of complex molecules containing n systems are obtained by computer programming based on approximate solutions of the Schrodinger wave equation. 

Molecular mechanics simulations are useful methods when dealing with large molecules or when limited information is required. When more sophisticated analysis is desired, such as thermodynamic data, it is usually necessary to switch to ab initio methods that seek to solve, or approximate a solution to, the Schrodinger wave equation for the entire molecule. Programs are rapidly improving both in terms of time taken to generate a solution to the molecular orbital and in the size of molecule that can be analyzed by these methods. Despite these advances only the simplest of supramolecular systems can usefully be investigated at this level of... 

Predicting the optical spectrum requires the solution of the Schrodinger wave equation in terms of the states ( a) and energy levels (Ea) of the system and the Hamiltonian... 

Statistical thermodynamics uses statistical arguments to develop a connection between the properties of individual molecules in a system and its bulk thermodynamic properties. For instance, consider a mole of water molecules at 25° C and standard pressure (1 bar). The thermodynamic state of the system has been defined on the basis of the number of molecules, the temperature, and the pressure. In order to relate the macroscopic thermodynamic properties such as U, G, H and A to the properties of the individual molecules, one would have to solve the Schrodinger wave equation (SWE) for a system composed of 6 x 10 interacting water molecules. This is an impossible task at present but if it were possible, one would obtain a wave function, I y, and an energy, 6)-, for the system. Moreover,... 

An ab initio method is a quantum-mechanical approach which attempts to calculate, from first principles, solutions to the Schrodinger wave equation. Molecular mechanics describes a system in which the energy depends only on the nuclei present. 

The equation is used to describe the behaviour of an atom or molecule in terms of its wave-like (or quantum) nature. By trying to solve the equation the energy levels of the system are calculated. However, the complex nature of multielectron/nuclei systems is simplified using the Born Oppenheimer approximation. Unfortunately it is not possible to obtain an exact solution of the Schrodinger wave equation except for the simplest case, i.e. hydrogen. Theoretical chemists have therefore established approaches to find approximate solutions to the wave equation. One such approach uses the Hartree Fock self-consistent field method, although other approaches are possible. Two important classes of calculation are based on ab initio or semi-empirical methods. Ab initio literally means from the beginning . The term is used in computational chemistry to describe computations which are not based upon any experimental data, but based purely on theoretical principles. This is not to say that this approach has no scientific basis — indeed the approach uses mathematical approximations to simplify, for example, a differential equation. In contrast, semi-empirical methods utilize some experimental data to simplify the calculations. As a consequence semi-empirical methods are more rapid than ab initio. 

The Schrodinger wave equation and its auxiliary postulates enable us to determine certain functions k of the coordinates of a system and the time. These functions are called the Schrodinger wave functions or probability amplitude functions. The square of the absolute value of a given wave function is interpreted as a probability distribution function for the coordinates of the system in the state represented by this wave function, as will be discussed in Section lOo. The wave equation has been given this name because it is a differential equation of the second order in the coordinates of the system, somewhat similar to the wave equation of classical theory. The similarity is not close, however, and we shall not utilize the analogy in our exposition. 

A wave function representing a one-electron system is then a function of four coordinates, three positional coordinates such as x, y, and z, and the spin coordinate . Thus we write p(x, y, z)a(u) and p(x, y, z)i3( ) as the two wave functions corresponding to the positional wave function p(x, y, z), which is a solution of the Schrodinger wave equation. The introduction of the spin wave functions for systems containing several electrons will be discussed later. 

Information about the wavefunction is obtained from the Schrodinger wave equation, which can be set up and solved either exactly or approximately the Schrodinger equation can be solved exactly only for a species containing a nucleus and only one electron (e.g. H, He ), i.e. a hydrogen-like system. 

By using the normal coordinates, the Schrodinger wave equation for the system can be written as... 

The simple de Broglie relation, X = hip, proved to be adequate to account for the properties of a particle moving with constant momentum in one dimension, but it cannot be applied to more complex systems where the momentum of the particle varies with position, or there is more than one variable. The equation which describes the wave behaviour of such systems is the celebrated Schrodinger wave equation, developed by Erwin Schrodinger in 1926. For one-dimensional systems with constant potential energy, this equation produces exactly the same results as those obtained from application of the de Broglie relation. 

In previous chapters we have considered systems for which there is an exact solution to the Schrodinger wave equation, but as we begin to look at atoms containing more than one electron we shall find that it is impossible to solve the Schrodinger equation exactly, and various approximations will have to be introduced to make the problem solvable. Before these are considered it will be useful to look at some basic concepts of quantum mechanics in more detail, so that we can obtain the Schrodinger equation for any system that may be of interest. 

The Schrodinger wave equation can only be used for a system with one electron, such as the hydrogen atom and cannot solve for larger atoms and molecules. Two approximations are used the valence bond and the molecular orbital methods. 

THE SCHRODINGER WAVE EQUATION In 1926, the Austrian physicist Erwin Schrodinger presented the equation relating the energy of a system to the wave motion. The Schrodinger equation is commonly written in the form... 

For our present purposes we shall take as our additional postulate the supposition that the mechanical behaviour of matter on the atomic scale is in accordance with the Schrodinger wave equation. Its numerical solution, for appropriate conditions, expresses the observable properties without contravening the principle that it is impossible to make an exact and simultaneous specification of position and velocities. It may be remarked that it is because of this principle of uncertainty that wave mechanics seek to describe the state of a system by means of the function whose purpose is to describe probabilities and not certainties. 

The discrete energy levels arise naturally as the allowed solutions of the wave equations for the system under consideration. Electronic energy levels in atoms may be accounted for by solving the Schrodinger wave equation. 

In principle, we can perform some sort of molecular orbital calculation on molecules of almost any complexity. It is, however, often extremely profitable to relate the properties of a complex system to those of a simpler one. Take, for example, the hydrogen atom in an electric field. It is much more instructive to see how the unperturbed levels of the atom are altered as a field is applied, than to solve the Schrodinger wave equation for the more complex case of the molecule with the field on. Analogously, to appreciate the orbital structure of complex systems it is much more insightful to start off with the levels of a simpler one and switch on a perturbation. 3.1-3.3 show three examples of different types of perturbations... 

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